It's starting to look like a pond!
We (unintentionally) timed it pretty much exactly right. The weather in the weeks we were digging it was unprecedentedly hot and dry for British May-time, and shortly after we finished it, the rain seems to have started. We are not using tap water at all, because, although the water around here is relatively soft, it's better overall that there are as few dissolved solids as possible for the health of any wildlife that may live in the water.
We started filling it with stored rainwater, but now that we're getting frequent rain, it doesn't seem necessary to artificially fill it. Looking at the weather forecast at the moment, I think it'll fill up itself in the next week.